The Frattini subgroup of a group $G$ is the intersection of the maximal subgroups of $G$, or $G$ itself in case $G$ has no maximal subgroups.

The Frattini subgroup is equal to the set of "non-generators" of $G$.  An element $g$ of $G$ is a non-generator if, whenever $G$ is generated by a set $S$ containing $g$, it is also generated by $S\setminus \left\{g\right\}$.

The Frattini subgroup of a finite group is nilpotent.

The FrattiniSubgroup( G ) command returns the Frattini subgroup of a group G.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{SmallGroup}\left(32\,5\right)\:$

$F≔\mathrm{FrattiniSubgroup}\left(G\right)$

$F≔\Phi \left(\mathrm{<\; a\; permutation\; group\; on\; 32\; letters\; with\; 5\; generators\; >}\right)$

$\mathrm{GroupOrder}\left(F\right)$

$8$

$\mathrm{IsNilpotent}\left(F\right)$

$\mathrm{true}$

$F≔\mathrm{FrattiniSubgroup}\left(\mathrm{DihedralGroup}\left(12\right)\right)$

$F≔\Phi \left({\mathbf{D}}_{12}\right)$

$\mathrm{GroupOrder}\left(F\right)$

$2$

$\mathrm{GroupOrder}\left(\mathrm{FrattiniSubgroup}\left(\mathrm{Alt}\left(4\right)\right)\right)$

$1$

$G≔\mathrm{QuasicyclicGroup}\left(7\right)$

$G≔{\mathbb{Z}}_{7^{\mathrm{\infty }}}$

$\mathrm{FrattiniSubgroup}\left(G\right)$

${\mathbb{Z}}_{7^{\mathrm{\infty }}}$

$F≔\mathrm{FrattiniSubgroup}\left(\mathrm{DirectProduct}\left(\mathrm{SemiDihedralGroup}\left(6\right)\&comma;\mathrm{GL}\left(2\&comma;3\right)\right)\right)$

$F≔\Phi \left(\mathrm{<\; a\; permutation\; group\; on\; 32\; letters\; with\; 4\; generators\; >}\right)$

$\mathrm{AreIsomorphic}\left(F\&comma;\mathrm{DirectProduct}\left(\mathrm{FrattiniSubgroup}\left(\mathrm{SemiDihedralGroup}\left(6\right)\right)\&comma;\mathrm{FrattiniSubgroup}\left(\mathrm{GL}\left(2\&comma;3\right)\right)\right)\right)$

$\mathrm{true}$

The GroupTheory[FrattiniSubgroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.